What is the general solution for (-5cot(x))/2+7/3 =-1/6 ?

The general solution for (-5cot(x))/2+7/3 =-1/6 is x= pi/4+pin

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Popular Trigonometry >

(-5cot(x))/2+7/3 =-1/6

Solution

\frac{- 5 cot ( x )}{2} + \frac{7}{3} = - \frac{1}{6}

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

\frac{- 5 cot ( x )}{2} + \frac{7}{3} = - \frac{1}{6}

Move

\frac{7}{3}

to the right side

\frac{- 5 cot ( x )}{2} + \frac{7}{3} = - \frac{1}{6}

Subtract

\frac{7}{3}

from both sides

\frac{- 5 cot ( x )}{2} + \frac{7}{3} - \frac{7}{3} = - \frac{1}{6} - \frac{7}{3}

Simplify

\frac{- 5 cot ( x )}{2} + \frac{7}{3} - \frac{7}{3} = - \frac{1}{6} - \frac{7}{3}

Simplify

\frac{- 5 cot ( x )}{2} + \frac{7}{3} - \frac{7}{3} : \frac{- 5 cot ( x )}{2}

\frac{- 5 cot ( x )}{2} + \frac{7}{3} - \frac{7}{3}

Add similar elements:

\frac{7}{3} - \frac{7}{3} = 0

= \frac{- 5 cot ( x )}{2}

Simplify

- \frac{1}{6} - \frac{7}{3} : - \frac{5}{2}

- \frac{1}{6} - \frac{7}{3}

Least Common Multiplier of

6, 3 : 6

6, 3

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

6

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{7}{3} :

multiply the denominator and numerator by

2

\frac{7}{3} = \frac{7 \cdot 2}{3 \cdot 2} = \frac{14}{6}

= - \frac{1}{6} - \frac{14}{6}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 1 - 14}{6}

Subtract the numbers:

- 1 - 14 = - 15

= \frac{- 15}{6}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{15}{6}

Cancel the common factor:

3

= - \frac{5}{2}

\frac{- 5 cot ( x )}{2} = - \frac{5}{2}

\frac{- 5 cot ( x )}{2} = - \frac{5}{2}

\frac{- 5 cot ( x )}{2} = - \frac{5}{2}

Multiply both sides by

2

\frac{- 5 cot ( x )}{2} = - \frac{5}{2}

Multiply both sides by

2

\frac{2 ( - 5 cot ( x ) )}{2} = 2 (- \frac{5}{2})

Simplify

\frac{2 ( - 5 cot ( x ) )}{2} = 2 (- \frac{5}{2})

Simplify

\frac{2 ( - 5 cot ( x ) )}{2} : - 5 cot (x)

\frac{2 ( - 5 cot ( x ) )}{2}

Remove parentheses:

(- a) = - a

= \frac{- 2 \cdot 5 cot ( x )}{2}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 10 cot ( x )}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{10 cot ( x )}{2}

Divide the numbers:

\frac{10}{2} = 5

= - 5 cot (x)

Simplify

2 (- \frac{5}{2}) : - 5

2 (- \frac{5}{2})

Remove parentheses:

(- a) = - a

= - 2 \cdot \frac{5}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{5 \cdot 2}{2}

Cancel the common factor:

2

= - 5

- 5 cot (x) = - 5

- 5 cot (x) = - 5

- 5 cot (x) = - 5

Divide both sides by

- 5

- 5 cot (x) = - 5

Divide both sides by

- 5

\frac{- 5 cot ( x )}{- 5} = \frac{- 5}{- 5}

Simplify

cot (x) = 1

cot (x) = 1

General solutions for

cot (x) = 1

cot (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (-5cot(x))/2+7/3 =-1/6 ?
The general solution for (-5cot(x))/2+7/3 =-1/6 is x= pi/4+pin